Distribution of first hitting time after a given date for Brownian Motion with Drift I have the following problem. Let $W$ be a standard brownian motion, let $D>0$, $B>0$ and $t^*>0$. I am interested in the distribution of the hitting time
$$\tau=\sup\{t\geq t^*>0: Dt+W(t)\geq A\}$$
Now, if this was $t^*=0$, I know how to find the distribution using the running maximum, however, in this case, the fact of being a time after a given date, is messing me up.
My Idea: It seems that I need to find a way of transforming the hitting time into a traditional first hitting time of a brownian motion starting with time zero, so I have to manipulate the brownian motion. In that direction, I can write
$$\mathbb{P}(\tau\leq x)=\mathbb{P}(\max_{t^*\leq t\leq x}(W(t)+Dt)\geq A)$$
And I'm tempted to start by adding and subtracting $W(t^*)$, but that would leave me with an additional random quantity in the expression. Any help or advice is welcome. Thanks!
 A: I think I found a Solution. Can anybody tell me if it is correct? It goes like this:
$\hat{W}(s)=W(t^*+s)-W(t^*)$ is a standard brownian motion from the Differential Propery, and it is independent of $W(t^*)$. With this observation, and letting $s=t-t^*$ for $t\geq s$, we can write, provided $x>t^*$,
$$\mathbb{P}(\tau\leq x)=\mathbb{P}(\max_{t^*\leq t\leq x}(W(t)+Dt)\geq A)=$$
$$\mathbb{P}(\max_{t^*\leq t\leq x}(W(t)-W(t^*)+Dt-Dt^*)\geq A-W(t^*)-Dt^*) $$
$$\mathbb{P}(\max_{0\leq s\leq x-t^*}(\hat{W}(s)+Ds)\geq A-W(t^*)-Dt^*)$$
So now we have a probability involving the running maximum of a standard brownian motion, with a random quantity on the Right Hand Side of the inequality in the last probability. As $W(t^*)$ is independent of $\hat{W}(s)$ for every $s$ however, we can finally write, letting $\phi_{0,t^*}$ be the $N(0,t^*)$ cdf,
$$\mathbb{P}(\tau\leq x)=\int_{-\infty}^\infty\mathbb{P}(\max_{0\leq s\leq x-t^*}(\hat{W}(s)+Ds)\geq A-w-Dt^*\mid W(t^*)=w)\phi_{0,t^*}(w)dw=$$
$$ \int_{-\infty}^\infty\mathbb{P}(\max_{0\leq s\leq x-t^*}(\hat{W}(s)+Ds)\geq A-w-Dt^*)\phi_{0,t^*}(w)dw$$
And $\mathbb{P}(\max_{0\leq s\leq x-t^*}(\hat{W}(s)+Ds)\geq A-w-Dt^*)$ can be computed using standard techniques for running maximum for Brownian Motion with Drift.
